Abstract

A method is described for the derivation of refractive properties and aberration structure of subapertures of freeform surfaces. Surface shapes are described in terms of Zernike polynomials. The method utilizes matrices to transform between Zernike and Taylor coefficients. Expression as a Taylor series facilitates the translation and size rescaling of subapertures of the surface. An example operation using a progressive addition lens surface illustrates the method.

© 2011 Optical Society of America

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References

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  1. C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
    [CrossRef]
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    [CrossRef]
  5. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
    [CrossRef]
  6. S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066(2006).
    [CrossRef]
  7. S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Optom. Vision Sci. 83, 666–671 (2006).
    [CrossRef]
  11. C. Fowler, “Recent trends in progressive power lenses,” Ophthalmic Physiol. Opt. 18, 234–237 (1998).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. E. A. Villegas and P. Artal, “Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions,” Optom. Vision Sci. 80, 106–114 (2003).
    [CrossRef]
  14. C. Zhou, W. Wang, K. Yang, X. Chai, and Q. Ren, “Measurement and comparison of the optical performance of an ophthalmic lens based on a Hartmann–Shack wavefront sensor in real viewing conditions,” Appl. Opt. 47, 6434–6441 (2008).
    [CrossRef] [PubMed]
  15. T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
    [CrossRef]
  16. P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.
  17. E. J. Sarver and M. T. Hall, “Fast evaluation of equal-spaced Zernike polynomial expansion samples,” J. Refract. Surg. 26, 61–65 (2010).
    [CrossRef] [PubMed]
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    [PubMed]
  19. American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (ANSI, 2004).
  20. L. W. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
    [PubMed]

2011 (1)

T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
[CrossRef]

2010 (1)

E. J. Sarver and M. T. Hall, “Fast evaluation of equal-spaced Zernike polynomial expansion samples,” J. Refract. Surg. 26, 61–65 (2010).
[CrossRef] [PubMed]

2009 (1)

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (4)

2004 (1)

American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (ANSI, 2004).

2003 (3)

C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
[CrossRef]

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

E. A. Villegas and P. Artal, “Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions,” Optom. Vision Sci. 80, 106–114 (2003).
[CrossRef]

2002 (3)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

2001 (2)

1998 (1)

C. Fowler, “Recent trends in progressive power lenses,” Ophthalmic Physiol. Opt. 18, 234–237 (1998).
[CrossRef] [PubMed]

1978 (1)

L. W. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

Alvarez, L. W.

L. W. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Ares, J.

Arines, J.

Artal, P.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Optom. Vision Sci. 83, 666–671 (2006).
[CrossRef]

E. A. Villegas and P. Artal, “Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions,” Optom. Vision Sci. 80, 106–114 (2003).
[CrossRef]

Bara, S.

Bastida, K.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Bianchetti, A.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Blendowske, R.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Optom. Vision Sci. 83, 666–671 (2006).
[CrossRef]

Campbell, C. E.

Chai, X.

Comastri, S.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Cox, I. G.

Dai, G. M.

Fowler, C.

C. Fowler, “Recent trends in progressive power lenses,” Ophthalmic Physiol. Opt. 18, 234–237 (1998).
[CrossRef] [PubMed]

Geary, K.

Goldberg, K. A.

Guirao, A.

Hall, M. T.

E. J. Sarver and M. T. Hall, “Fast evaluation of equal-spaced Zernike polynomial expansion samples,” J. Refract. Surg. 26, 61–65 (2010).
[CrossRef] [PubMed]

Lundstrom, L.

Martin, G.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Pankretz, G. S.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.

Perez, G.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Perez, L.

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

Prado, P.

Raasch, T. W.

T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
[CrossRef]

Ren, Q.

Riera, P. R.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.

Sarver, E. J.

E. J. Sarver and M. T. Hall, “Fast evaluation of equal-spaced Zernike polynomial expansion samples,” J. Refract. Surg. 26, 61–65 (2010).
[CrossRef] [PubMed]

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Su, L.

T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
[CrossRef]

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Topa, D. M.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.

Unsbo, P.

Villegas, E. A.

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Optom. Vision Sci. 83, 666–671 (2006).
[CrossRef]

E. A. Villegas and P. Artal, “Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions,” Optom. Vision Sci. 80, 106–114 (2003).
[CrossRef]

Wang, W.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Williams, D. R.

Yang, K.

Yi, A.

T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
[CrossRef]

Zhou, C.

Appl. Opt. (1)

J. Am. Optom. Assoc. (1)

L. W. Alvarez, “Development of variable-focus lenses and a new refractor,” J. Am. Optom. Assoc. 49, 24–29 (1978).
[PubMed]

J. Opt. A (1)

S. Comastri, K. Bastida, A. Bianchetti, L. Perez, G. Perez, and G. Martin, “Zernike aberrations when pupil varies: selection rules, missing modes and graphical method to identify modes,” J. Opt. A 11, 085302 (2009).
[CrossRef]

J. Opt. Soc. Am. A (8)

J. Refract. Surg. (2)

E. J. Sarver and M. T. Hall, “Fast evaluation of equal-spaced Zernike polynomial expansion samples,” J. Refract. Surg. 26, 61–65 (2010).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” J. Refract. Surg. 18, S652–660 (2002).
[PubMed]

Ophthalmic Physiol. Opt. (1)

C. Fowler, “Recent trends in progressive power lenses,” Ophthalmic Physiol. Opt. 18, 234–237 (1998).
[CrossRef] [PubMed]

Optom. Vision Sci. (4)

A. Guirao and D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

E. A. Villegas and P. Artal, “Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions,” Optom. Vision Sci. 80, 106–114 (2003).
[CrossRef]

T. W. Raasch, L. Su, and A. Yi, “Whole-surface characterization of progressive addition lenses,” Optom. Vision Sci. 88, E217–226 (2011).
[CrossRef]

R. Blendowske, E. A. Villegas, and P. Artal, “An analytical model describing aberrations in the progression corridor of progressive addition lenses,” Optom. Vision Sci. 83, 666–671 (2006).
[CrossRef]

Other (2)

American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (ANSI, 2004).

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” in Optical Design and Analysis Software II, R.C.Juergens, ed. (SPIE, 2002), pp. 130–144.

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Figures (3)

Fig. 1
Fig. 1

Surface height of freeform surface. Diameter of 60 mm , contour lines at 50 μm intervals.

Fig. 2
Fig. 2

Dioptric powers of surface. Contour lines in 0.25 D intervals, assumed refractive index = 1.50 . (a) Spherical power of surface. Dioptric power range 2.53 D. (b) Astigmatic magnitude of surface. Dioptric range from 0.01 D to 1.44 D. (c) Astigmatic component with horizontal/vertical principal meridia. Dioptric range from -1.18 D to +0.53 D. (d) Astigmatic component with diagonal principal meridia. Dioptric range from -1.01 D to +1.01 D.

Fig. 3
Fig. 3

High-order aberrations of surface. RMS error magnitude ( μm ). Surface shapes within 4 mm diameter subapertures. Color range ± 0.20 μm .

Tables (1)

Tables Icon

Table 1 Zernike Terms and Coefficients ( μ m ) up through the Fifth Order for a Freeform Surface of 60 mm Diameter and Zernike Coefficients for a 4 mm Pupil Positioned at Five Locations over That Lens a

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ZT ( x , y ) = ( 1 2 y 2 x 2 6 x y 3 ( 2 x 2 + 2 y 2 1 ) 6 ( x 2 y 2 ) 2 2 ( 3 x 2 y y 3 ) 3 2 ( x 8 28 x 6 y 2 + 70 x 4 y 4 28 x 2 y 6 + y 8 ) 0 2 0 2 6 x 4 3 y 2 6 y 2 2 ( 3 x 2 3 y 2 ) 24 2 ( 7 x 6 y 35 x 4 y 3 + 21 x 2 y 5 y 7 ) 0 0 2 2 6 y 4 3 x 2 6 x 12 2 x y 24 2 ( x 7 21 x 5 y 2 + 35 x 3 y 4 7 x y 6 ) 0 0 0 0 2 3 6 6 2 y 84 2 ( x 6 15 x 4 y 2 + 15 x 2 y 4 y 6 ) 0 0 0 2 6 0 0 12 2 x 336 2 ( 3 x 5 y 10 x 3 y 3 + 3 x y 5 ) 0 0 0 0 2 3 6 6 2 y 84 2 ( x 6 15 x 4 y 2 + 15 x 2 y 4 y 6 ) 0 0 0 0 0 0 2 2 168 2 ( 5 x 4 y 10 x 2 y 3 + y 5 ) 0 0 0 0 0 0 0 3 2 ) .
T j = T j ( r 2 / r 1 ) n j ,
TZ = [ ZT ( 0 , 0 ) ] 1 ,
TZ = ( 1 0 0 1 / 4 0 1 / 4 0 7 / 128 0 1 / 2 0 0 0 0 1 / 4 0 0 0 1 / 2 0 0 0 0 0 0 0 0 0 6 / 12 0 0 0 0 0 0 3 / 12 0 3 / 12 0 7 3 / 192 0 0 0 6 / 12 0 6 / 12 0 7 6 / 192 0 0 0 0 0 0 2 / 16 0 0 0 0 0 0 0 0 2 / 768 ) .
c c = TZ × ( ( ZT ( x , y ) × c c ) . * r s j ) ,

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